# Specializations of one-parameter families of polynomials

Farshid Hajir^{[1]}; Siman Wong^{[1]}

- [1] University of Massachusetts Department of Mathematics & Statistics Amherst, MA 01003-9318 (USA)

Annales de l’institut Fourier (2006)

- Volume: 56, Issue: 4, page 1127-1163
- ISSN: 0373-0956

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topHajir, Farshid, and Wong, Siman. "Specializations of one-parameter families of polynomials." Annales de l’institut Fourier 56.4 (2006): 1127-1163. <http://eudml.org/doc/10168>.

@article{Hajir2006,

abstract = {Let $K$ be a number field, and suppose $ \lambda (x,t)\in K[x, t] $ is irreducible over $K(t)$. Using algebraic geometry and group theory, we describe conditions under which the $K$-exceptional set of $\lambda $, i.e. the set of $\alpha \in K$ for which the specialized polynomial $\lambda (x,\alpha )$ is $K$-reducible, is finite. We give three applications of the methods we develop. First, we show that for any fixed $n\ge 10$, all but finitely many $K$-specializations of the degree $n$ generalized Laguerre polynomial $ L_n^\{(t)\}(x) $ are $K$-irreducible and have Galois group $S_n$. Second, we study specializations of the modular polynomial $\Phi _n(x,t)$ (which vanishes on the $j$-invariants of pairs of elliptic curves related by a cyclic $n$-isogeny), and show that for any $ n\ge 53 $, all but finitely many of the $K$-specializations of $ \Phi _n(x, t) $ are $K$-irreducible and have Galois group containing $\{\rm SL\}_2(\mathbb\{Z\}/n)/\lbrace \pm I \rbrace $. Third, for a simple branched cover $\pi :Y\rightarrow \mathbb\{P\}_K^\{1\}$ of degree $n\ge 7$ and of genus at least $2$, all but finitely many $K$-specializations are $K$-irreducible and have Galois group $S_n$.},

affiliation = {University of Massachusetts Department of Mathematics & Statistics Amherst, MA 01003-9318 (USA); University of Massachusetts Department of Mathematics & Statistics Amherst, MA 01003-9318 (USA)},

author = {Hajir, Farshid, Wong, Siman},

journal = {Annales de l’institut Fourier},

keywords = {Branched cover; complex multiplication; Hilbert irreducibility; modular equation; orthogonal polynomial; rational point; Riemann-Hurwitz formula; simple cover; specialization; branched cover},

language = {eng},

number = {4},

pages = {1127-1163},

publisher = {Association des Annales de l’institut Fourier},

title = {Specializations of one-parameter families of polynomials},

url = {http://eudml.org/doc/10168},

volume = {56},

year = {2006},

}

TY - JOUR

AU - Hajir, Farshid

AU - Wong, Siman

TI - Specializations of one-parameter families of polynomials

JO - Annales de l’institut Fourier

PY - 2006

PB - Association des Annales de l’institut Fourier

VL - 56

IS - 4

SP - 1127

EP - 1163

AB - Let $K$ be a number field, and suppose $ \lambda (x,t)\in K[x, t] $ is irreducible over $K(t)$. Using algebraic geometry and group theory, we describe conditions under which the $K$-exceptional set of $\lambda $, i.e. the set of $\alpha \in K$ for which the specialized polynomial $\lambda (x,\alpha )$ is $K$-reducible, is finite. We give three applications of the methods we develop. First, we show that for any fixed $n\ge 10$, all but finitely many $K$-specializations of the degree $n$ generalized Laguerre polynomial $ L_n^{(t)}(x) $ are $K$-irreducible and have Galois group $S_n$. Second, we study specializations of the modular polynomial $\Phi _n(x,t)$ (which vanishes on the $j$-invariants of pairs of elliptic curves related by a cyclic $n$-isogeny), and show that for any $ n\ge 53 $, all but finitely many of the $K$-specializations of $ \Phi _n(x, t) $ are $K$-irreducible and have Galois group containing ${\rm SL}_2(\mathbb{Z}/n)/\lbrace \pm I \rbrace $. Third, for a simple branched cover $\pi :Y\rightarrow \mathbb{P}_K^{1}$ of degree $n\ge 7$ and of genus at least $2$, all but finitely many $K$-specializations are $K$-irreducible and have Galois group $S_n$.

LA - eng

KW - Branched cover; complex multiplication; Hilbert irreducibility; modular equation; orthogonal polynomial; rational point; Riemann-Hurwitz formula; simple cover; specialization; branched cover

UR - http://eudml.org/doc/10168

ER -

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## Citations in EuDML Documents

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